Concavity and inflection points of f(x) = -x^3 + 3x^2 + 4 Step by Step

Step by step

The question

Question: find the concavity and the inflection points of f(x) = −x³ + 3x² + 4.

f(x) = −x³ + 3x² + 4

Concave up means the curve bends up, like a cup.

That is where f″ is positive.

Concave down bends down, like a cap: f″ is negative.

cup concave up: f″ > 0

cap concave down: f″ < 0

1 Find f″(x)

First f′: the power rule on each term.

f′(x) = −3x^(3 − 1) + 3 × 2x^(2 − 1) + 0

Now we have f′(x) = −3x² + 6x.

= −3x² + 6x

Then f″: the power rule again, on f′.

f″(x) = −3 × 2x^(2 − 1) + 6 × 1

Now we have f″(x) = −6x + 6.

= −6x + 6

2 Solve f″(x) = 0

The bending can only change where f″ is 0.

So we solve f″(x) = 0.

f″(x) = 0

−6x + 6 = 0

Take out the −6.

−6(x − 1) = 0

So x = 1.

x = 1

3 Sign table for f″

A sign table for f″.

The number 1 cuts the line into 2 parts.

Test one number in each part: 0, 2.

f″(0) = 6, f″(2) = −6

f″(0) is 6: positive.

So f bends up there: a cup.

cup

f″(2) is −6: negative.

So f bends down there: a cap.

cap

Careful!

Concave up is about the bending, not about going up.

A cup can go down on one side.

Careful: concave up is not "going up"

4 Inflection points

At x = 1, the bending changes: an inflection point.

infl.

Its y: put 1 into f.

We get 6.

f(1) = −1³ + 3(1)² + 4 = −1 + 3 + 4 = 6

f is concave up where f″ is positive: x < 1.

up ∪ : x < 1

f is concave down where f″ is negative: x > 1.

down ∩ : x > 1

And the inflection point is (1, 6).

inflection : (1, 6)

5 Check

Let's check with the graph.

A cup, then a cap.

It changes at the inflection point.

Right!

Let's look back.

First: find f″(x).

Then: solve f″(x) = 0.

Then: sign table for f″.

Then: inflection points.

Last: check.

Answer